HW4soln

HW4soln - CE 573: Structural Dynamics Solutions to HW#4...

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CE 573: Structural Dynamics Solutions to HW#4 Problem #1 Given : A response spectrum is a plot of the maximum dynamic load factor (DLF) as a function of the (nondimensional) duration of the loading. See Figures 4.4 and 4.5 in the text for examples. We wish to find the response spectrum of the triangular pulse loading shown above. Assume that the dynamic system that this loading is applied to is undamped with a mass of 1 kg and a spring stiffness of 2 4 π N/m. (Hence, the period of free vibration is T = 1 second.) Also, assume that the forcing function has a maximum value of 2 4 N. Required : a) Obtain a plot of the response spectrum for this loading as follows: 1) Using any appropriate numerical method, obtain plots of the response of the dynamic system to this loading for the following values of : 0.05 seconds, 0.06 seconds, 0.07 seconds, 0.08 seconds, 0.09 seconds, 0.10 seconds, 0.15 seconds, and 0.20 seconds. (You do d t not need to turn in printouts of the actual calculations – just the plots.) 2) For each value of , determine the maximum value of the response and identify this value on your plots. d t 3) Plot the maximum response values versus . (You may wish to plot them as shown in Figures 4.4 and 4.5.) d t b) Compare the values you obtained for your response spectrum with the values of the spectrum shown in Figure 4.5 (for a different type of triangular loading). What can you say about the influence of the shape of the pulse on the response spectrum values for these two cases? Solution : I will use the linear acceleration method to solve this problem – the set-up of the method is shown below: 1. Need to pick a time step and check its stability. The linear acceleration method requires for stability; any will have good accuracy. I will choose 0.551 tT ∆≤ /10 t =0.005 seconds. 2. Preliminary Calculations – 2 0.0 m; 0.0 m/sm; ( 0) 0 N. 0 m/s . ooo oo Fc uk u o m uu F t u −− == = = ⇒= = ± ± ±± 3. Linear Acceleration Method: () ( ) 2 23 63 N mm 2 61 kg 30 4 240.04 10 . 0.005 sec 0.005 sec mc ii t t kk =+ + = + + = × N
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I will show the calculations for one of the loading cases: 0.05 d t = seconds. The other cases will be handled similarly. Linear Acceleration Method Mass Damping Stiffness Time Step Effective Stiffness (kg) (kg/s) (N/m) (sec) (N/m) 1.00 0 39.4780 0.005 240040.00 t_i F_i u_i udot_i udotdot_i D Fbar_i D u_i D udot_i (sec) (N) (m) (m/s) (m/s^2) (N) (m) (m/s) 0.000 0.0000 0.0000 0.0000 0.0000 7.8957 3.289E-05 1.974E-02 0.005 7.8957 0.0000 0.0197 7.8944 55.2621 2.302E-04 5.919E-02 0.010 15.7914 0.0003 0.0789 15.7810 149.9477 6.247E-04 9.858E-02 0.015 23.6870 0.0009 0.1775 23.6520 291.8585 1.216E-03 1.379E-01 0.020 31.5827 0.0021 0.3154 31.4997 480.8541 2.003E-03 1.770E-01 0.025 39.4780 0.0041 0.4924 39.3159 700.9556 2.920E-03 1.765E-01 0.030 31.5827 0.0070 0.6690 31.3053 888.7823 3.703E-03 1.364E-01 0.035 23.6870 0.0107 0.8054 23.2634 1028.3571 4.284E-03 9.615E-02 0.040 15.7914 0.0150 0.9015 15.1987 1119.5424 4.664E-03 5.579E-02 0.045 7.8957 0.0197 0.9573 7.1189 1162.2483 4.842E-03 1.537E-02 0.050 0.0000 0.0245 0.9727 -0.9680 1164.3285 4.851E-03 -5.325E-03 0.055 0.0000 0.0294 0.9674 -1.1595 1157.3640 4.822E-03 -6.280E-03 0.060 0.0000 0.0342 0.9611 -1.3498 1149.2575 4.788E-03 -7.228E-03 0.065 0.0000 0.0390 0.9539 -1.5388 1140.0170 4.749E-03 -8.169E-03 0.070 0.0000 0.0437 0.9457 -1.7263 1129.6516 4.706E-03 -9.102E-03 0.075 0.0000 0.0484 0.9366 -1.9121 1118.1715 4.658E-03 -1.003E-02 0.080 0.0000 0.0531 0.9266 -2.0960 1105.5882 4.606E-03 -1.094E-02 0.085 0.0000 0.0577 0.9156 -2.2778 1091.9139 4.549E-03 -1.184E-02 0.090 0.0000 0.0622 0.9038 -2.4574 1077.1622 4.487E-03 -1.274E-02 0.095 0.0000 0.0667 0.8910 -2.6346 1061.3477 4.422E-03 -1.362E-02 0.100 0.0000 0.0712 0.8774 -2.8091 1044.4860 4.351E-03 -1.448E-02 0.105 0.0000 0.0755 0.8629 -2.9809 1026.5937 4.277E-03 -1.533E-02 0.110 0.0000 0.0798 0.8476 -3.1498 1007.6885 4.198E-03 -1.617E-02 0.115 0.0000 0.0840 0.8314 -3.3155 987.7890 4.115E-03 -1.699E-02 0.120 0.0000 0.0881 0.8145 -3.4779 966.9150 4.028E-03 -1.779E-02 0.125 0.0000 0.0921 0.7967 -3.6370 945.0869 3.937E-03 -1.858E-02 0.130 0.0000 0.0961 0.7781 -3.7924 922.3264 3.842E-03 -1.935E-02 0.135 0.0000 0.0999 0.7587 -3.9441 898.6559 3.744E-03 -2.009E-02 0.140 0.0000 0.1036 0.7386 -4.0919 874.0988 3.641E-03 -2.082E-02 0.145 0.0000 0.1073 0.7178 -4.2356 848.6792 3.536E-03 -2.153E-02 0.150 0.0000 0.1108 0.6963 -4.3752 822.4224 3.426E-03 -2.222E-02 0.155 0.0000 0.1143 0.6741 -4.5105 795.3542 3.313E-03 -2.288E-02 0.160 0.0000 0.1176 0.6512 -4.6413 767.5013 3.197E-03 -2.353E-02 0.165 0.0000 0.1208 0.6277 -4.7675 738.8913 3.078E-03 -2.415E-02 0.170 0.0000 0.1238 0.6035 -4.8890 709.5523 2.956E-03 -2.474E-02 0.175 0.0000 0.1268 0.5788 -5.0057 679.5133 2.831E-03 -2.531E-02
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HW4soln - CE 573: Structural Dynamics Solutions to HW#4...

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